# Toffoli gate evolution

For the sake of simplicity, I'm dropping unnecessary constants like $\hbar$. I'm working on a problem in quantum control theory, and trying to find a test case to synthesize the ternary Toffoli gate. Unfortunately, I cannot seem to find a basis that admits a possible solution.

In order to remove the control theoretical aspect from this problem and make it only quantum mechanical, if (X,Y,Z) are the two-level Pauli spin matrices, I have the following basis vectors

$$B= \left\{ I \otimes I \otimes X, I \otimes I \otimes Y, I \otimes X \otimes I, I \otimes Y \otimes I, X \otimes I \otimes I, Y \otimes I \otimes I, Z \otimes Z \otimes Z \right\}$$

and I'm trying to create the Toffoli gate

$$T=\begin{pmatrix} 1 \\ & 1 \\ && 1 \\ &&& 1 \\ &&&&1 \\ &&&&&1 \\ &&&&&& 0 & 1 \\ &&&&&& 1 & 0 \end{pmatrix}$$

Now according to numerical calculations, the basis set spans a 72-dimensional Lie group(which seems wrong since it should only span a 64 dimensional space). In any case, the problem seems to be that there's no way to express the Toffoli gate as a discrete time evolution of the identity operator using elements of the basis set, that is

$$T = e^{i H_n t_n} e^{i H_{n-1} t_{n-1}} \cdots e^{i H_1 t_1} I$$

where $t_i > 0, H_i \in B$ for all i=1,...,n.

So to sum up, I have numerically computed a seemingly inconsistent dimension of the Lie Group, and in any case it seems that despite having a suitable quorum to construct the Toffoli gate, I am unable to. Any thoughts?

$$T = e^{i H_n t_n} e^{i H_{n-1} t_{n-1}} \cdots e^{i H_1 t_1} I$$
I think the Hamiltonian is $$|11\rangle\langle 11|\otimes (\sigma_x - I)$$